Hey Buddy..You Got Change For A Dollar?
Another math problem:
Using just pennies, nickels, dimes, and quarters, how many distinct ways are there to get $1 (order does not matter).
The (sometimes) incoherent and inconsistent Ramblings of a math geek, chess addict, and father of two young and adorable kids.
Another math problem:
Using just pennies, nickels, dimes, and quarters, how many distinct ways are there to get $1 (order does not matter).
Tags: math
3 comments:
You're really good at making me doubt my ability to do math!
Anyway, I can't see how to solve this algebraically. I can see that the equation:
25Q + 10D + 5N + P = 100,
where Q = # of quarters, D = # of dimes, N = # of nickels and P = # of pennies
represents the set of answers, as long as we limit the variables to being greater than or equal to 0 and integer only.
I then end up looking at brute force approaches, such as:
Consider looking for ways to get a total of 25, and then calculate the product of those combos in getting 100.
One way to get 25 is with Q alone. All other combos assume no Q's.
If D = 2 (its highest possible value), then your only choices are N = 1 & P = 0, or N = 0 & P = 5. So that's 2 more combinations for getting to 25, bringing us to 3 so far.
If D = 1, then your choices are N = 3 & P = 0, N = 2 & P = 5, N = 1 & P = 15, and N = 0 & P = 15. That's 4 more combinations, for a total of 7 so far in getting to 25.
If D = 0, then your choices are N = 5 & P = 0, N = 4 & P = 5, N = 3 & P = 10, N = 2 & P = 15, N = 1 & P = 20, and N = 0 & P = 25. That's 6 more combinations, for a total of 13 so far.
So now we know there are 13 combinations of Q, D, N & P that uniquely add up to 25.
To get to 50, each combination of the 13 options would be represented by 13 * 13, or 169. However, that misses one combination that wouldn't have been arrived at from combining the original 13 with itself, which is D = 5. So the adjusted total is 170.
To get from 50 to 100, do the same cross-combining of the answer for 50, or 170 * 170, for a total of 28,900 combinations of Q, D, N & P to get to 100.
No idea if this is right, but I can't see anything that I've missed, so that's my answer!
One of my "& P = 15" should have been "& P = 25", of course. But it didn't affect the answer (just my notes on how I got to it)!
Hey Math Geek, how come you didn't celebrate the fact that this was your 64th blog entry? That's 2 to the power 6, you know!
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